Notes 2.1 - Limits

Inverse FunctionsNotes 3.8 I. Inverse FunctionsA.)

B.) Ex.

C.)

D.) Symmetric to the line y = x.

E.) Notation

F.) Existence: A function has an inverse iff for any two x values

Horizontal Line Test for Inverses

II. Inverse TheoremsA.)

B.)

C.) Ex. GivenDoes it have an inverse, and if so, what is it?

Always positive, therefore always increasing!Cannot solve for y!D.) Derivatives of Inverse Functions:

andE.) Ex- Given inverse functions

and

Notice

F.) Ex- Given inverse functions

and

G.) Ex- Given

1.) Does it have an inverse?

2.) If it does, find it and then find its derivative.3.) Verify the inverse derivative theorem on

Always positive, therefore always increasing, and it has an inverse

You verify (f (5), 5)!! H.)

Notice, f (3) = 9. The most confusing aspect of the inverse derivative theorem is that you are asked to find the derivative at a value of x. You are really being asked to find the derivative of the inverse function at the value that corresponds to f (x) = 9.

I.)

J.)

Derivatives of Inverse Trig FunctionsNotes 3.8 Part III. y = sin-1 x

II. y = cos-1 x

III. y = tan-1 x

IV. y = cot-1 x

V. y = sec-1 x

VI. y = csc-1 x

Find y in each of the following:

VII. Examples